Marginal Win Value and Dollars Per Win

It is quite natural to spawn opinions on offseason moves that eventually manifest into conclusions that the general manager of Team A is a moron for signing Player B to Contract X. No matter our qualifications or level of knowledge of baseball, there are multiple times per winter us fans are left with the notion that yes, we absolutely have to be smarter than the GM of Team A.

What we do not take into consideration is that within every front office in Major League Baseball, there are dozens of bright analytical and scouting minds who are far more conversant than us. Their scouts can see how one player’s bat speed may be diminishing. Their analytical department knows how a free agent’s skill set will fit in their ballpark far better than we do. Their doctors have medical reports on players that we could only dream of seeing. The information accessible to these bright minds is far more than we can find on MLB.TV, Fangraphs, and Baseball-Reference. Essentially, front offices make educated decisions dependent on a wealth of knowledge that the public lacks.

There are a multitude of factors that drive every transaction. The Mariners did not invest $240 million dollars in Robinson Cano because they thought he was a pretty damn good second baseman. Inside that decision came scouting insight, statistical analysis, and economic evaluation that indicated it was sensible for the Mariners to blow every single team in the market out of the water.

Of course, there are rudimentary differences in each front office’s valuation of every player. If each team valued every player equally, the market would be far different. Some teams put more weight into the scouting aspect of evaluations, while others rely on regression models to project players forward. Another large factor that is not touched on nearly enough is the marginal value that each player provides his employer.

For a number of years, analysts have looked at the contracts signed by free agents, quantified the value of the free agent, then depicted how much money the market demanded for a single Win Above Replacement ($/WAR). This number has risen annually for the past decade for reasons such as inflation and an influx of TV money among others. The number for this off-season’s contracts was generally pegged at $6 to $7 million dollars per win. That is to say that if a team is going to sign a player who has consistently posted 2 WAR seasons (league-average), the market values him at around $13 million per season. These numbers are pretty consistent with how the market has played out so far.

However, the money per win valuation has numerous caveats that must be considered. First of all, $/WAR is not a good way to project salaries among elite players. As godly as he is, it’s unlikely that the 10-WAR man Mike Trout would be paid $70 million per year if he were on the open market. Robinson Cano posted a 6 WAR season in 2013, meaning if the market always held true, he would have found a contract that paid him an AAV of around $40 million dollars. The second caveat is that while WAR is an extremely useful and powerful tool to evaluate players, front offices look far beyond that metric while putting a dollar sign on the value of a player. Another caveat, one which I find most interesting, is that a win is worth a different amount to every team. This in a nutshell is marginal win value.

The basic concept of marginal win value goes like this: a team filled with AAA players wins about 47 or so games in the major leagues while costing around $12 million dollars. From 47 wins on, the team is buying wins in hopes of accruing more revenue from being good. However, if the team buys one win at the market rate of $7 million dollars, the revenue added from increasing their win total from 47 to 48 is worth far less than $7 million dollars. From 47 wins on, it takes marginal analysis to decide if buying more wins is profitable for the team. A single win carries far different value for teams who are on opposite sides of the spectrum. Several extremely smart people such as Phil Birnbaum and Nate Silver have looked at the marginal win value curve, creating the following graph to depict its true worth.

(Click to enlarge)

This graph is from 2005, so it is fairly outdated. In 2005, Silver found a win to cost around $2 million on the open market. To adjust to inflated contracts and revenue, we would simply raise the graph’s Y value to equate it to today’s contracts, keeping the shape consistent. The graph shows that it is an inefficient allocation of money to keep buying wins from around 60-85. This makes sense because barring a team slipping in one of the wildcard slots, it’s unlikely a team makes the playoffs with 85 wins. However, the theory states that at 86 wins, the value of the revenue a team makes from extra ticket sales or whatever revenues are driven by playoff excitement begins to exceed the money you are paying for a win. Looking at the graph, you can see that you acquire surplus value buying wins from 85-95 because the marginal value is higher than the $2 million dollar cost of a win in 2005. From 95 on, the value decreases due to the fact that you likely have the division title in the bag and spending money to win upwards of 100 games is rather gluttonous.

There are uncertainties when using marginal win value to dictate your offseason. It is extremely hard to project how many wins a player is worth, especially with the volatility of pitchers or injury-risk players. That, along with other possible variations, makes it extremely difficult for a team to forecast its win totals while entering a season. The standard deviation between true talent and performance is about six wins in the best projection models, meaning that a team with the talent of an 81 win team could win anywhere from 75 to 87 games. That being said, marginal win value is absolutely something that front offices take into account when deciding if it is profitable to keep spending on their club.

In Part 2, I am going to take a look at some of the most questionable moves that were made this offseason, and try to find each front office’s motives behind the moves. While marginal win value is going to play a large role in this exercise, I’m also going to use a number of other explanations to decipher why these transactions were made.

A lot of this piece was made possible by the fantastic work of people such as Dave Cameron, Lewie Pollis, Phil Birnbaum, and Nate Silver. These articles are fantastic reads and quite informative. Special thanks to Matt Swartz for the correspondence and help in equating Silver’s 2005 graph to the inflated contracts of this offseason.

About AuthorDaniel Schoenfeld

Daniel Schoenfeld is a senior in high school in Evanston, Illinois. He plans on studying business economics while pitching in college next year. He is interested in scouting and statistical analysis, and hopes to use Batting Leadoff as a platform to break into the industry. He’s currently working on a large scale project detailing the indicative factors involved in injury projection of pitchers and is always willing to learn and share. Contact him at dschon711@aol.com and follow him @DanielSchoe.

The author did not prove that Cano is NOT being paid $40 million for this year. Who says his long-term contract is for equal amounts every year? The last year of his contract could be viewed as a $6 million year.

In accounting, there’s the ‘cash method’ and the ‘accrual method’. The accrual method is the more sophisticated. In baseball the equivalent would be to apportion the long-term contract in terms of expected WAR each year. For Cano, it might be 5 WAR expected in the first year, 3 in the next few years, and then 1 for the remaining years, with 0 or -1 in the final year.

A more realistic ‘cost curve’ would put fans’ hatred of overpaid decliners into perspective. ARod shouldn’t be hated as much for his second-half-of-contract disintegration. Pujols and Hamilton should be hated much more for failing to deliver in the highest-paid year of the contract: the first year.

Another way to view a long-term contract would be to project total WAR over the length of the contract, and adjust for expected inflation.

The absence of a $70 million per year man can be explained by agents’ preference for a larger dollar figure over more years.

Just because a huge one-year contract hasn’t happened yet, doesn’t mean that dollars per WAR calculations are nonsensical.